The Study Of Shock Diffraction Problem In Multi-dimensional Space

The present Ph.D dissertation is devoted to the study of the global existence, stability and regularity of the solutions for the shock diffraction problem by wedge, modeled by2-d nonlinear wave system and potential flow equations respectively. Hyperbolic conservation laws have broad applications in real life, whose typical ex-ample is Euler system. As the important system in gas dynamics, Euler system describes the inviscid fluid motion in space. In order to deal with real problems, researchers often make some assumptions on some properties for convenience, and then study the deduced simpler models. The nonlinear wave system or potential flow equations for instance. In general, a planar shock wave will interact with wedge. Noticing that these systems are invariant under uniform stretching of space-time co-ordinates and if the initial data is invariant under stretching of space coordinates, we can use self-similar coordinates to reduce the number of coordinate. In self-similar coordinates, both nonlinear wave system and potential flow equations can be trans-formed into second order nonlinear partial differential equations. These equations are hyperbolic in supersonic region, while elliptic in subsonic region. The problems of shock diffraction by wedge is then deduced to study the nonlinear partial differ-ential equations of mixed type with free boundary. To study the global existence and uniform regularity is the main part of this dissertation. The new ideas and new methods developed here is also applied to the problems involved similar difficulties. The study of these problems is of practical significance and of theoretical value, since there are many important applications in geometry and physics, etc., for nonlinear partial differential equations of mixed type.The dissertation is organized as follows.Chapter One is an introduction. It is devoted to introducing physical back- ground and previous mathematical research works on shock reflection and diffrac-tion. The main problems, main results, and methods in this Ph.D. dissertation are also illustrated.In Chapter Two, it deals with the shock diffraction to the convex wedge for the flow modeled by nonlinear wave system, and the global existence and optimal regularity near degenerate line is obtained.In Chapter Three, it continues to deal with the shock diffraction to the convex wedge for the flow modeled by potential flow equations. Compared to nonlinear wave system, there is no symmetry with respect to wedge angle, and the derivatives of velocity potential should be estimated for potential flow equation. On the other hand, the angle exterior to the wedge corner is bigger than π, it is hard to obtain the derivative estimates for elliptic equations, which is the crucial issue for this problem. Thus new ideas and more precise estimates is needed. Finally, the global existence and optimal regularity near degenerate line is obtained.